方差【标准差】
标准差公式是一种数学公式。标准差也被称为标准偏差，或者实验标准差，公式如下所示：
样本标准差=方差的算术平方根=s=sqrt(((x1-x)^2 +(x2-x)^2 +......(xn-x)^2)/n)
总体标准差=σ=sqrt(((x1-x)^2 +(x2-x)^2 +......(xn-x)^2)/n )
注解：上述两个标准差公式里的x为一组数（n个数据）的算术平均值。当所有数（个数为n）概率性地出现时（对应的n个概率数值和为1），则x为该组数的数学期望。
公式意义
所有数（个数为n）记为一个数组[n]。将数组的所有数求和后除以n得到算术平均值。数组的所有数分别减去平均值，得到的n个差值分别取平方，再将得到的所有平方数求和，然后除以数的个数或个数减一若所求为总体标准差则除以n,若所求为样本标准差则除以(n-1)，最后把得到的商取算术平方根，就是取1/2次方，得到的结果就是这组数(n个数据)的标准差。
新建窗体应用程序AverageAndDeviationDemo，将默认的Form1重命名为FormAverageAndDeviation。
窗体FormAverageAndDeviation的视图编辑器源程序如下：
FormAverageAndDeviation.Designer.cs
namespace AverageAndDeviationDemo
{
partial class FormAverageAndDeviation
{
/// <summary>
/// 必需的设计器变量。
/// </summary>
private System.ComponentModel.IContainer components = null;
/// <summary>
/// 清理所有正在使用的资源。
/// </summary>
/// <param name="disposing">如果应释放托管资源，为 true；否则为 false。</param>
protected override void Dispose(bool disposing)
{
if (disposing && (components != null))
{
components.Dispose();
}
base.Dispose(disposing);
}
#region Windows 窗体设计器生成的代码
/// <summary>
/// 设计器支持所需的方法 - 不要修改
/// 使用代码编辑器修改此方法的内容。
/// </summary>
private void InitializeComponent()
{
this.rtxtMessage = new System.Windows.Forms.RichTextBox();
this.btnGenerate = new System.Windows.Forms.Button();
this.label1 = new System.Windows.Forms.Label();
this.SuspendLayout();
//
// rtxtMessage
//
this.rtxtMessage.Location = new System.Drawing.Point(217, 95);
this.rtxtMessage.Name = "rtxtMessage";
this.rtxtMessage.Size = new System.Drawing.Size(687, 550);
this.rtxtMessage.TabIndex = 0;
this.rtxtMessage.Text = "";
//
// btnGenerate
//
this.btnGenerate.Location = new System.Drawing.Point(38, 93);
this.btnGenerate.Name = "btnGenerate";
this.btnGenerate.Size = new System.Drawing.Size(149, 23);
this.btnGenerate.TabIndex = 1;
this.btnGenerate.Text = "生成随机集合后开始比较";
this.btnGenerate.UseVisualStyleBackColor = true;
this.btnGenerate.Click += new System.EventHandler(this.btnGenerate_Click);
//
// label1
//
this.label1.AutoSize = true;
this.label1.Location = new System.Drawing.Point(215, 64);
this.label1.Name = "label1";
this.label1.Size = new System.Drawing.Size(143, 12);
this.label1.TabIndex = 2;
this.label1.Text = "迭代Loop与Linq性能比较:";
//
// FormAverageAndDeviation
//
this.AutoScaleDimensions = new System.Drawing.SizeF(6F, 12F);
this.AutoScaleMode = System.Windows.Forms.AutoScaleMode.Font;
this.ClientSize = new System.Drawing.Size(981, 714);
this.Controls.Add(this.label1);
this.Controls.Add(this.btnGenerate);
this.Controls.Add(this.rtxtMessage);
this.Name = "FormAverageAndDeviation";
this.Text = "计算集合平均值和标准差【方差】的示例";
this.Load += new System.EventHandler(this.FormAverageAndDeviation_Load);
this.ResumeLayout(false);
this.PerformLayout();
}
#endregion
private System.Windows.Forms.RichTextBox rtxtMessage;
private System.Windows.Forms.Button btnGenerate;
private System.Windows.Forms.Label label1;
}
}
窗体代码如下：
FormAverageAndDeviation.cs
using System;
using System.Collections.Generic;
using System.ComponentModel;
using System.Data;
using System.Diagnostics;
using System.Drawing;
using System.Linq;
using System.Text;
using System.Threading.Tasks;
using System.Windows.Forms;
namespace AverageAndDeviationDemo
{
public partial class FormAverageAndDeviation : Form
{
public FormAverageAndDeviation()
{
InitializeComponent();
rtxtMessage.ReadOnly = true;
}
/// <summary>
/// 显示操作内容
/// </summary>
/// <param name="text"></param>
private void DisplayMessage(string text)
{
this.BeginInvoke(new Action(() =>
{
if (rtxtMessage.TextLength > 20480)
{
rtxtMessage.Clear();
}
rtxtMessage.AppendText($"{DateTime.Now.ToString("HH:mm:ss.fff")}->{text}\n");
rtxtMessage.ScrollToCaret();
}));
}
private void FormAverageAndDeviation_Load(object sender, EventArgs e)
{
List<double> array = new List<double>() { 1, 2, 3, 4, 5, 6 };
CalculateAverageAndDeviation(array);
CalculateAverageAndDeviationUseLoop(array);
}
/// <summary>
/// 计算 平均值和 标准差【方差】
/// 使用Linq迭代查询
/// </summary>
/// <param name="array"></param>
private void CalculateAverageAndDeviation(IList<double> array)
{
if (array == null || array.Count == 0)
{
throw new ArgumentException($"无法对空集合求取平均值");
}
Stopwatch stopwatch = new Stopwatch();
stopwatch.Start();
//获取 平均值
double average = array.Average();
//获取方差之和
double sumDeviation = array.Select(x => Math.Pow(x - average, 2)).Sum();
//目标 标准差
double standardDeviation = Math.Sqrt(sumDeviation / array.Count);
stopwatch.Stop();
double elapsedTime = stopwatch.Elapsed.TotalMilliseconds;//耗时ms
DisplayMessage($"【Linq】average:{average}\nsumDeviation:{sumDeviation}\nstandardDeviation:{standardDeviation}\nElapsedTime:{elapsedTime}ms");
}
/// <summary>
/// 计算 平均值和 标准差【方差】
/// 使用循环迭代器
/// </summary>
/// <param name="array"></param>
private void CalculateAverageAndDeviationUseLoop(IList<double> array)
{
if (array == null || array.Count == 0)
{
throw new ArgumentException($"无法对空集合求取平均值");
}
Stopwatch stopwatch = new Stopwatch();
stopwatch.Start();
double sum = 0;
for (int i = 0; i < array.Count; i++)
{
sum += array[i];
}
//获取 平均值
double average = sum / array.Count;
sum = 0;
for (int i = 0; i < array.Count; i++)
{
sum += Math.Pow(array[i] - average, 2);
}
//获取方差之和
double sumDeviation = sum;
//目标 标准差
double standardDeviation = Math.Sqrt(sumDeviation / array.Count);
stopwatch.Stop();
double elapsedTime = stopwatch.Elapsed.TotalMilliseconds;//耗时ms
DisplayMessage($"【Loop】average:{average}\nsumDeviation:{sumDeviation}\nstandardDeviation:{standardDeviation}\nElapsedTime:{elapsedTime}ms");
}
/// <summary>
/// 生成10.0~99.9之间的随机集合
/// </summary>
/// <param name="count"></param>
/// <returns></returns>
private double[] GenerateRandomArray(int count)
{
double[] array = new double[count];
for (int i = 0; i < count; i++)
{
array[i] = new Random(i + 1).Next(100, 1000) * 0.1;
}
return array;
}
private async void btnGenerate_Click(object sender, EventArgs e)
{
double[] array = GenerateRandomArray(500);
DisplayMessage($"打印源数组【{string.Join(",", array)}】");
Task linqTask = Task.Run(() => CalculateAverageAndDeviation(array));
Task loopTask = Task.Run(() => CalculateAverageAndDeviationUseLoop(array));
Task.WaitAll(linqTask, loopTask);
await loopTask;
}
}
}
测试运行如图：
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